Non Color Data Procedural Normals
Overview:

I think that everything should be ok with just Bump node. There is a method to convert Height Map (B&amp;W) to the same Normal Map as in your first image, but this produce the same results.

Comparison:

First my node setup:




  Bump_to_normal group is not mine, see Credits at the bottom.


In Texture Group Node there is Height Map created from Voronoi Texture.

Bump node:



Bump_to_normal node:



As you can see there is no difference at all, no matter from what map you feed Normal input in Glass node.

Solution:

I think your problem lays just in Distance value in Bump node. It's to high.

About Bump to Normal:

I can't tell you how this node setup is working, there is little to none info from the author (check Credits). But I'll show you how to set it up:




In Texture Node Group > Height Map Image set your Height map. It could be any B&amp;W input. Image Texture, Texture Paint, Procedural, you name it.
Connect Bump_to_normal Node Group to Normal Map and then just to any Normal input you want.


Blend file:



Credits:

First of all this is not my work, it's from riceart41 on BlendSwap (download this file to make all of this to work). You can find more information on YouTube and on BlenderArtists.